Asymptotic Behavior of Underlying NT Paths in Interior Point
Method for Monotone Semidefinite Linear Complementarity
Problems
Chee-Khian Sim1 2
Communicated by F. Potra
1This work was partially supported by a startup grant from The Hong Kong Polytechnic University. Part of
this research was done while the author is a research fellow at the Logistics Institute - Asia Pacific, Singapore,
and at the NUS Business School. The author is grateful to Prof. Paul Tseng for starting him on the proof of
Proposition 3.2, and in getting Example 3.1.2Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong
Kong. Email: [email protected]
This is the Pre-Published Version.
Abstract
An interior point method (IPM) defines a search direction at each interior point of the feasible region.
These search directions form a direction field, which in turn gives rise to a system of ordinary differential
equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the
system of ODEs. In [32], these off-central paths are shown to be well-defined analytic curves and any
of their accumulation points is a solution to the given monotone semidefinite linear complementarity
problem (SDLCP). In [32]-[34], the asymptotic behavior of off-central paths corresponding to the HKM
direction is studied. In particular, in [32], the authors study the asymptotic behavior of these paths
for a simple example, while, in [33,34], the asymptotic behavior of these paths for a general SDLCP
is studied. In this paper, we study off-central paths corresponding to another well-known direction,
the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-
central paths to be analytic w.r.t.√
µ and then w.r.t. µ, at solutions of a general SDLCP. Also, as in
[32], we present off-central path examples using the same SDP, whose first derivatives are likely to be
unbounded as they approach the solution of the SDP. We work under the assumption that the given
SDLCP satisfies a strict complementarity condition.
Keywords: Semidefinite linear complementarity problem; Interior point methods; NT direction; Local
convergence; Ordinary differential equations
AMS Subject Classification: 90C51; 90C22; 34C05
1 Introduction.
The notion of a central path is introduced by Sonnevend [35] in 1985 to interior point methods (IPMs).
Since then, researchers realize that an IPM is actually a homotopy method following underlying paths
and that many remarkable properties of an IPM are attributed to the nice geometry of these paths.
Readers who are interested to know more about basic geometry of these paths may refer to [2].
An important role the underlying paths play in the study of IPMs is to show its fast local conver-
gence. The classical proof of local convergence of an iterative method, such as the Newton’s method,
for finding the solution of a system of equations relies on the nonsingularity of the Jacobian matrix.
However, the Jacobian matrix of the equation system, defining the search direction in an IPM, may be
singular at an optimal solution. Thus, the traditional approach of local convergence analysis does not
work for IPM. Study of underlying paths, which mimic the behavior of iterates generated by an interior
point algorithm, especially when the iterates are close together at a solution of the given problem, pro-
vides an alternative approach. Fast local convergence of IPMs has been successfully proved by relating
it to the boundedness of derivatives of the underlying paths in [27,38,42,43]. See also [36,37]. Superlin-
ear convergence of IPMs for linear complementarity problem (LCP) is proved in [18,19,20,14,6,24] by
explicitly using the analyticity of off-central path. Also, [10,9,21,22,23] contain results about superlinear
convergence of IPMs for SDP.
Study of fast local convergence is particularly important for the class of monotone semidefinite
linear complementarity problems (SDLCPs), with the class of semidefinite programs (SDPs) as a special
case, because, in contrast to a monotone linear complementarity problem (LCP), the exact solution of
a SDLCP cannot be obtained from an approximate solution by determining a complementary basis.
The analysis for SDP, and hence for SDLCP, is considered to be more difficult than that for linear
programming (LP). This arises mainly due to the difficulty in maintaining symmetry in the linearized
complementarity [44]. Researchers working in the IPM area have proposed ways to overcome this
drawback, which result in different symmetrized search directions, along which iterates generated by
interior point algorithms move [1,7,13,28,29,30,31,41]. Among these search directions, the AHO, HKM
and NT directions are more commonly used.
There are various ways in which the underlying paths, using these different search directions, for
SDLCPs are defined in literature [3,15,17,26,32,40]. Paths arising from different search directions are
likely to behave differently from each other. In [32], a new definition of the underlying paths of IPMs
for SDLCPs, using ordinary differential equations (ODEs), is proposed. The motivation for defining
paths in this way is to relate these paths to the vector field of search directions of an IPM. In [32]-
[34], the behavior of these off-central paths, corresponding to the HKM direction, near solutions of a
1
SDLCP is investigated. It is found that, for off-central paths corresponding to the HKM direction,
their asymptotic behavior [32] suggests that superlinear convergence of iterates, generated by a generic
interior point algorithm using the HKM direction, may not be guaranteed. This contrasts with that for
interior point algorithms using the AHO direction, where it has been shown that superlinear convergence
of iterates, generated by a generic interior point algorithm (which does not, say, perform “narrowing” of
the neighborhood), is possible [1,12,16,17,25,26]. It is interesting to investigate the asymptotic behavior
of off-central paths corresponding to another well-known direction, the so-called NT direction [30,31],
to see whether they behave like that corresponding to the AHO direction or that corresponding to the
HKM direction.
In this paper, we use our definition of paths for SDLCPs [32] to study the asymptotic behavior of
SDLCP paths, corresponding to the NT direction. To the author’s knowledge, this study is the first to
study off-central paths corresponding to the NT direction. In Section 2, through the same simple SDP
example that we used in [32], we show the behavior of a few off-central paths as they approach the
unique solution of the SDP. We show graphically that except for the central path, the first derivatives
of all the other paths are likely to be unbounded as these paths approach the unique solution of the
SDP. In Section 3, considering a general SDLCP, we give necessary and sufficient conditions for an
off-central path, corresponding to the NT direction, to be analytic in the limit, w.r.t.√
µ and then
w.r.t. µ. Here, µ is related to the variable in the primal space and the variable in the dual space for a
SDP (see Remark 2.1). Finally, we give some concluding remarks in Section 4.
1.1 Notations and Common Definitions.
The space of symmetric n×n matrices is denoted by Sn. Given matrices X and Y in <p×q, the standard
inner product is defined by X • Y ≡ Tr(XT Y ), where Tr(·) denotes the trace of a matrix. If X ∈ Sn
is positive semidefinite (resp., positive definite), we write X º 0 (resp., X Â 0). The cone of positive
semidefinite (resp., positive definite) symmetric matrices is denoted by Sn+ (resp., Sn
++). Either the
identity matrix or operator will be denoted by I. Eij ∈ <n×n is a square matrix with 1 at its (i, j)
entry and the rest of entries equal to zero. ‖ · ‖ for a vector in <n refers to its Euclidean norm, and for
a matrix in <p×q, it refers to its Frobenius norm.
For a matrix X ∈ <p×q, we denote its component at the ith row and jth column by Xij . Also,
Xi· denotes the ith row of X, and X·j the jth column of X. In case X is partitioned into blocks of
submatrices, then Xij refers to the submatrix in the corresponding (i, j) position.
Given square matrices Ai ∈ <ni×ni , i = 1, . . . , m, Diag(A1, . . . , Am) is a square matrix with Ai as
its diagonal blocks arranged in accordance to the way they are lined up in Diag(A1, . . . , Am). All the
2
other entries in Diag(A1, . . . , Am) are taken to be zero.
Given functions f : Ω −→ E and g : Ω −→ <++, where Ω is an arbitrary set and E is a
normed vector space, and a subset Ω ⊆ Ω, we write f(w) = O(g(w)) for all w ∈ Ω to mean that
‖f(w)‖ ≤ Mg(w) for all w ∈ Ω, where M > 0 is a positive constant. Suppose we have E = Sn. Then
we write f(w) = Θ(g(w)) if for all w ∈ Ω, f(w) ∈ Sn++, f(w) = O(g(w)) and f(w)−1 = O(1/g(w)).
What the subset Ω is should be clear from the context. Usually, Ω = (0, w) for a small w > 0.
A function f = (f1, . . . , fm) from an open subset O of <k to <m is analytic at a point x =
(x1, . . . , xk) ∈ O iff each fi, i = 1, . . . , m, can be written as a convergent power series expansion about
x in an open neighborhood of x. Furthermore, if x0 ∈ <k is on the boundary of O, we say f is analytic
at x0 (or can be extended analytically to x0), and we let f(x0) = limx→x0 f(x), iff there exists an
analytic function g, which is analytic at x0 and coincides with f wherever both are defined.
The above also applies if a component xj of x = (x1, . . . , xk) is a symmetric matrix, instead of a
number, in which case, we consider xj to lie in an Euclidean space of appropriate dimension. If the
range of f is in the space of (symmetric) matrices, we also view the space as an appropriate Euclidean
space, when considering analyticity, so that the above applies.
2 Definition of an Off-central Path and an Investigation on NT
Paths for an Example.
Let us consider the following SDLCP:
XY = 0, (1)
A(X) + B(Y ) = q, (2)
X, Y ∈ Sn+, (3)
where A,B : Sn −→ <n are linear operators mapping Sn to the space <n, where n := n(n+1)/2. Hence
A and B have the form A(X) = (A1 • X, . . . , An • X)T , respectively, B(Y ) = (B1 • Y, . . . , Bn • Y )T ,
where Ai, Bi ∈ Sn for all i = 1, . . . , n. Also, q ∈ <n.
We have the following assumptions on the SDLCP throughout the paper:
Assumption 2.1 (a) SDLCP is monotone, i.e., A(X) + B(Y ) = 0 for X, Y ∈ Sn ⇒ X • Y ≥ 0.
(b) There exist X1, Y 1 Â 0, such that A(X1) + B(Y 1) = q.
(c) A(X) + B(Y ) : X, Y ∈ Sn = <n.
3
Assumption 2.1(a)-(c) are basic assumptions used in the literature when SDLCP is studied in the
context of IPM.
Let us now define the off-central path for SDLCP passing through a point (X0, Y 0), X0, Y 0 Â 0,
satisfying A(X) + B(Y ) = q.
Definition 2.1 The solution (X(µ), Y (µ)), X(µ), Y (µ) Â 0, to
HP (XY ′ + X ′Y ) =1µ
HP (XY ), (4)
A(X ′) + B(Y ′) = 0, (5)
where µ > 0, with the initial condition (X(1), Y (1)) = (X0, Y 0), X0, Y 0 Â 0, is the off-central path
for SDLCP, corresponding to P , passing through (X0, Y 0). Here HP (U) := 12 (PUP−1 + (PUP−1)T ),
and P ∈ <n×n is an invertible matrix.
Assuming P be an analytic function of X, Y and PXY P−1 be always symmetric (such P include
the well-known directions like the HKM, and its dual, and NT directions), it is proved in [32] that the
above is well-defined, and (X(µ), Y (µ)) is unique, analytic over µ ∈ (0,∞). The motivation for defining
an off-central path as in Definition 2.1 is also given in [32].
Remark 2.1 The central path (Xc(µ), Yc(µ)) for SDLCP, which satisfies Xc(µ)Yc(µ) = µI, is a special,
but important, example of an off-central path for SDLCP. When µ = 1, it satisfies Tr(Xc(1)Yc(1)) = n.
Therefore, we also require the initial data (X0, Y 0) when µ = 1 in (4)-(5) to satisfy Tr(X0Y 0) = n. In
this case, using (4), it is easy to see that the parameter µ in the ODE system (4)-(5) actually represents
the duality gap, X(µ) • Y (µ), at the point (X(µ), Y (µ)) on the path.
Let us now consider a SDP example as follows:
(P) min
0 0
0 1
•X
subject to
2 0
0 0
•X = 2,
0 −1
−1 2
•X = 0, X ∈ S2
+
and
(D) max 2v1
subject to v1
2 0
0 0
+ v2
0 −1
−1 2
+ Y =
0 0
0 1
, Y ∈ S2
+.
This example, which first appeared in [11], has all the nice properties (e.g. primal and dual
nondegeneracy, strict complementarity, e.t.c.) for a SDP. The example is also used in [?] to analyze the
asymptotic behavior of its off-central paths corresponding to the HKM direction.
4
It has an unique solution,
1 0
0 0
,
0 0
0 1
, which satisfies strict complementarity and
nondegeneracy.
Written as a SDLCP, the above example can be expressed as
XY = 0
Asvec(X) + Bsvec(Y ) = q
X, Y ∈ S2+,
where A =
2 0 0
0 0 0
0 −√2 2
, B =
0 0 0
0√
2 1
0 0 0
and q =
2
1
0
. Note that A and B are the
corresponding matrix representation of the linear operator A and B respectively in (2).
In [32], off-central paths for the example are analyzed. The off-central paths analyzed correspond to
the dual HKM direction, which are obtained by setting P = Y 1/2 in (4). It is shown analytically in [32]
that unless an off-central path satisfies certain algebraic condition, its first derivatives are unbounded
in the limit as the path approaches the unique solution of the example. In this section, we consider
off-central paths for the same SDP example, but corresponding to the NT direction, where P in (4) is
such that PT P = W−1, W is a symmetric, positive definite matrix with WY W = X.
Although it is known that one can express W in terms of X and Y , it is still difficult to write
down explicitly each entry of W in terms of that of X and Y for the example, even if these matrices
are only of size two. We are therefore unable to further analyze the asymptotic behavior of these paths
analytically, like the (dual) HKM case in [32]. As an alternative, we have written Matlab codes to study
the behavior of the first derivatives of various off-central paths corresponding to the NT direction, for
the example.
Using Tr(X0Y 0) = 2 and (5), we have the off-central path for the example, (X(µ), Y (µ)), passing
through (X0, Y 0) = (X(1), Y (1)), is of the form
X(µ) =
1 2µ− y1(µ)
2µ− y1(µ) 2µ− y1(µ)
and Y(µ) =
y1(µ) y2(µ)
y2(µ) 1− 2y2(µ)
.
The central path for the example is given by
Xc(µ) =
1 2µ− yc
1(µ)
2µ− yc1(µ) 2µ− yc
1(µ)
and Yc(µ) =
yc
1(µ) yc2(µ)
yc2(µ) 1− 2yc
2(µ)
,
where yc2(µ) = −yc
1(µ) and
yc1(µ) =
2µ√1 + 4µ2 + 1− 2µ
. (6)
5
In the below plots, X0 are all set to be equal to Xc(1), while Y 0 are varied by varying y02 = y2(1),
keeping y01 = y1(1) equal to yc
1(1) = 2√5−1
.
2 2.5 3 3.5 4 4.5 5 5.5
x 10−10
300
310
320
330
340
350
360
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
y20=0.25
2 2.5 3 3.5 4 4.5 5 5.5
x 10−10
1150
1200
1250
1300
1350
1400
1450
1500
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
y20=0
2 2.5 3 3.5 4 4.5 5 5.5
x 10−10
5000
5500
6000
6500
7000
7500
8000
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
y20=−0.6
1 1.5 2 2.5 3 3.5
x 10−5
1.99
2
2.01
2.02
2.03
2.04
2.05
2.06
2.07
2.08
2.09
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
y20=−1.6
1 1.5 2 2.5 3 3.5
x 10−5
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1624
3.1624
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
Central Path
6
2 2.5 3 3.5 4 4.5 5 5.5
x 10−10
450
500
550
600
650
700
750
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
y20=−1.6
2 2.5 3 3.5 4 4.5 5 5.5
x 10−10
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
3.1623
µ
Su
m o
f n
orm
sq
ua
red
of firs
t d
eriva
tive
s
Central Path
As observed from these plots, we see that except for the case when the off-central path is actually
the central path, the sums of norm squared of first derivatives1 of the given paths get larger as µ
approaches zero. The first three plots are for paths that are initiated relatively far from the central
path, while the fourth and sixth plots are for the same off-cental path that is initiated near to the
central path. Comparison between the fourth and fifth plot shows that for this off-central path with
y02 = −1.6, for the same range of µ, the sum of norm squared of its derivatives starts to increase while
that of the central path remains stable. For the case of the central path, one can observe that its sum
of first derivatives’ norm squared is approaching 3.162 . . . as µ approaches zero (seventh plot), which
should be the case, as can be verified using yc2(µ) = −yc
1(µ) and (6).
Hence, from these graphical examples, they indicate the possibility that unless the off-central paths
corresponding to the NT direction, for the example, satisfy certain condition, their first derivatives are
likely to be unbounded as µ tends to zero.
3 Asymptotic Analyticity Behavior of NT Paths for a general
SDLCP.
Based on what is observed of the first derivatives of off-central paths for the SDP example in the above
section, it suggests that off-central paths corresponding to the NT direction are in general not analytic
at solutions of a SDLCP. In this section, we study the conditions that are required to ensure analyticity
of paths corresponding to the NT direction, for a general SDLCP. We investigate the asymptotic
analyticity of an off-central path, corresponding to the NT direction, first w.r.t.√
µ and then w.r.t.
1Sum of norm squared of first derivatives here stands for√‖X′(µ)‖2 + ‖Y ′(µ)‖2.
7
µ. By studying the conditions which ensure asymptotic analyticity of paths, one can hope to ensure
that iterates generated by interior point algorithms are guaranteed to converge superlinearly as they
approach a solution of a given SDLCP.
We require an additional assumption, besides Assumption 2.1, to carry out the analysis.
Assumption 3.1 There exists a strictly complementary solution, (X∗, Y ∗), to SDLCP (1)-(3).
The analysis of the asymptotic behavior of an off-central path for a general SDLCP is considered
to be difficult without this assumption (Assumption 3.1). However, we note that there have been some
work done in this area for special classes of SDLCP without the assumption. See for example [5].
Let (X∗, Y ∗) be a strictly complementary solution to SDLCP (1)-(3), which exists by Assumption
3.1.
Since X∗ and Y ∗ commute, they are jointly diagonalizable by some orthogonal matrix. So, using
a suitable orthogonal similarity transformation of the matrices in SDLCP (1)-(3), we may assume,
without loss of generality, that
X∗ =
Λ∗11 0
0 0
, Y ∗ =
0 0
0 Λ∗22
,
where Λ∗11 = Diag(λ∗1, . . . , λ∗m) Â 0 and Λ∗22 = Diag(λ∗m+1, . . . , λ
∗n) Â 0. Here λ∗1, . . . , λ
∗n are real
numbers greater than zero.
Hereafter, whenever we partition a matrix S ∈ Sn, we do it in a similar way, i.e., S is always
partitioned as
S11 S12
ST12 S22
, where S11 ∈ Sm, S22 ∈ Sn−m and S12 ∈ <m×(n−m). The same holds for
S ∈ <n×n, which may not necessarily be symmetric.
3.1 Analyticity w.r.t.√
µ.
Let us now analyze the analyticity of off-central paths, corresponding to the NT direction, w.r.t.√
µ.
From now onwards, we may occasionally suppress the dependence of a variable on another variable.
For example, instead of writing X(µ) for the primal part of an off-central path, we simply write X. We
do this only when the dependency is clear from the context, and also for the sake of not cluttering an
expression with too many symbols.
Written in matrix-vector form, the ODE system (4)-(5) can be expressed as:
svec(A1)T svec(B1)T
......
svec(An)T svec(Bn)T
P ⊗s (P−T Y ) (PX)⊗s P−T
svec(X ′)
svec(Y ′)
=
1µ
0
svec(HP (XY ))
, (7)
8
where n = n(n + 1)/2.
Here, the operation ⊗s and the map svec (with inverse smat) are used, whose properties are given
on pp. 775− 776 and the appendix of [39].
Considering the NT direction, which corresponds to P , such that PT P = W−1, W a symmetric,
positive definite matrix with WY W = X, (7) becomes
svec(A1)T svec(B1)T
......
svec(An)T svec(Bn)T
I W ⊗s W
svec(X ′)
svec(Y ′)
=
1µ
0
svec(X)
, (8)
by taking P = W−1/2 in (7).
Remark 3.1 Note that W can be explicitly expressed in terms of X, Y by
W = X1/2(X1/2Y X1/2)−1/2X1/2 = Y −1/2(Y 1/2XY 1/2)1/2Y −1/2,
using WY W = X, X, Y,W symmetric, positive definite matrices.
Following the approach in [33], we analyze the asymptotic analyticity behavior of an off-central
path, (X(µ), Y (µ)), w.r.t.√
µ, by performing a transformation on (8) to an equivalent ODE system so
that analysis is possible.
Let X1(t) := X(t2), Y1(t) := Y (t2). Define X1(t), Y1(t) by
X1(t) =
I 0
0 tI
X1(t)
I 0
0 tI
(9)
and
Y1(t) =
tI 0
0 I
Y1(t)
tI 0
0 I
. (10)
Now W (µ) for an off-central path, (X(µ), Y (µ)), corresponding to the NT direction, is such that
WY W = X. From this, we have
W
tI 0
0 I
Y1
tI 0
0 I
W =
I 0
0 tI
X1
I 0
0 tI
using (9) and (10). This implies that
1t
tI 0
0 I
W
tI 0
0 I
Y1
tI 0
0 I
W
tI 0
0 I
1
t= X1.
9
Define
W1(t) :=1t
tI 0
0 I
W (t2)
tI 0
0 I
. (11)
Remark 3.2 X1(t), Y1(t) and W1(t) (of an off-central path) given above in (9)-(11) satisfy W1Y1W1 =
X1. Hence, we can write
W1 = X1/21 (X1/2
1 Y1X1/21 )−1/2X
1/21 (12)
by Remark 3.1. Therefore, W1 is an analytic function of X1, Y1, and for (X1(t), Y1(t)) corresponding
to the off-central path (X(µ), Y (µ)), W1(t) is an analytic function of t. Also, by (9), and that
X1(t) =
Θ(1) O(t)
O(t) Θ(t2)
,
we have X1(t) is symmetric, positive definite in the limit as t approaches zero. Similar property holds
for Y1(t). Using (12), W1(t) is hence symmetric, positive definite in the limit as t approaches zero.
Let us now reformulate (8) in terms of X1(t), Y1(t), W1(t), their derivatives and t, in order to
analyze the asymptotic analyticity behavior of (X1(t), Y1(t)) = (X(t2), Y (t2)), as t approaches zero.
From (8), we have
svec(X ′) + (W ⊗s W )svec(Y ′) =1µ
svec(X).
Expressing the above in terms of X1(t), Y1(t), W1(t), their derivatives and t(=√
µ), we obtain
svec(X ′1) + (W1 ⊗s W1)svec(Y ′
1)
=1t
I 0
0 −I
⊗s I
svec(X1)− 1
t(W1 ⊗s W1)
I 0
0 −I
⊗s I
svec(Y1).
(13)
Following [33],
svec(A1)T
...
svec(An)T
svec(X ′) +
svec(B1)T
...
svec(Bn)T
svec(Y ′) = 0
from (8) can be expressed as
A(t)svec(X ′1) + B(t)svec(Y ′
1) = −G(t)svec(X1)−H(t)svec(Y1), (14)
10
where
A(t)k· :=
svec
(Ak)11 t(Ak)12
t(Ak)T12 t2(Ak)22
T
for 1 ≤ k ≤ i1
svec
0 (Ak)12
(Ak)T12 t(Ak)22
T
for i1 + 1 ≤ k ≤ i1 + i2
svec
0 0
0 (Ak)22
T
for i1 + i2 + 1 ≤ k ≤ n
, (15)
B(t)k· :=
svec
t2(Bk)11 t(Bk)12
t(Bk)T12 (Bk)22
T
for 1 ≤ k ≤ i1
svec
t(Bk)11 (Bk)12
(Bk)T12 0
T
for i1 + 1 ≤ k ≤ i1 + i2
svec
(Bk)11 0
0 0
T
for i1 + i2 + 1 ≤ k ≤ n
, (16)
G(t)k· :=
svec
0 (Ak)12
(Ak)T12 2t(Ak)22
T
for 1 ≤ k ≤ i1
svec
0 0
0 (Ak)22
T
for i1 + 1 ≤ k ≤ i1 + i2
0 for i1 + i2 + 1 ≤ k ≤ n
, (17)
and
H(t)k· :=
svec
2t(Bk)11 (Bk)12
(Bk)T12 0
T
for 1 ≤ k ≤ i1
svec
(Bk)11 0
0 0
T
for i1 + 1 ≤ k ≤ i1 + i2
0 for i1 + i2 + 1 ≤ k ≤ n
, (18)
for some fixed i1, i2.
Details on how A(t),B(t),G(t) and H(t) are derived can be obtained from [33].
Combining (13) and (14), we have that (X1(t), Y1(t)), corresponding to the off-central path
11
(X(µ), Y (µ)), satisfies A(t) B(t)
I W1 ⊗s W1
svec(X ′
1)
svec(Y ′1)
=
−G(t) −H(t)
1t
I 0
0 −I
⊗s I − 1
t (W1 ⊗s W1)
I 0
0 −I
⊗s I
svec(X1)
svec(Y1)
.
(19)
We have the following proposition:
Proposition 3.1 A(t) B(t)
I W1 ⊗s W1
which appears on the left hand side of (19) is analytic and invertible for all t ≥ 0 and W1 positive
definite.
Proof: The analyticity of A(t) B(t)
I W1 ⊗s W1
is clear.
To show that it is invertible for all t ≥ 0 and W1 positive definite, it suffices to show that A(t) B(t)
I W1 ⊗s W1
u
v
= 0 ⇒ u = v = 0.
Now A(t) B(t)
I W1 ⊗s W1
u
v
= 0
implies that
A(t)u + B(t)v = 0
u + (W1 ⊗s W1)v = 0. (20)
A(t)u + B(t)v = 0 ⇒ uT v ≥ 0, a proof of which can be found from the proof of Proposition 2.3 in [33].
By positive definite of W1, and (20), we then have u = v = 0. QED
Consider
I 0
0 −I
⊗s I
svec(X1)− (W1 ⊗s W1)
I 0
0 −I
⊗s I
svec(Y1)
12
in (19).
Expanding the above expression and in matrix form, it is equal to (X1)11 0
0 −(X1)22
− W1
(Y1)11 0
0 −(Y1)22
W1. (21)
Using (9)-(11), (21) and Proposition 3.1 on the ODE system (19), we have the following theorem:
Theorem 3.1 Let (X(µ), Y (µ)), µ > 0, be an off-central path for SDLCP (1)-(3) under Assumptions
2.1 and 3.1. Then X(µ), Y (µ) are analytic as functions of t =√
µ at t = 0 if and only if
1t
I 0
0 1t I
X11(µ) 0
0 −X22(µ)
−W (µ)
Y11(µ) 0
0 −Y22(µ)
W (µ)
I 0
0 1t I
is analytic as a function of t =√
µ at t = 0.
A sufficient condition for analyticity of (X(µ), Y (µ)) as a function of t =√
µ is given in the
following:
Theorem 3.2 Let (X(µ), Y (µ)), µ > 0, be an off-central path for SDLCP (1)-(3) under Assumptions
2.1 and 3.1. If W12(µ) converges to zero as µ tends to zero, and is analytic as a function of t =√
µ at
t = 0, then X(µ), Y (µ) are analytic as functions of t =√
µ at t = 0.
Proof. Suppose that W12(µ) converges to zero as µ tends to zero, and is analytic as a function of t =√
µ
at t = 0. From (11), this is equivalent to (W1)12(t) converges to zero as t → 0, and is analytic at t = 0,
since (W1)12(t) = W12(t2).
Given X1, Y1, W1, symmetric, positive definite, with W1Y1W1 = X1. It is easy to show that each block
component of the matrix in (21) can be expressed in terms of (W1)12 (or its transpose), other block
entries of W1 and block entries of X1, Y1.
Now, (W1)12(t) = tV (t), where V (t) is analytic at t = 0. Hence, the matrix in (21) can be expressed as
a product of t and an analytic matrix function of (t, X1, Y1).
Therefore, (X1(t), Y1(t)) of the given off-central path satisfies an ODE system, in standard form (y′ =
f(t, y)), derived from (19), whose right hand side is analytic at any accumulation points of (X1(t), Y1(t)),
as t → 0. Hence, by Theorem 4.1 of [4], pp. 15 and Theorem 2.1 of [32], we have (X1(t), Y1(t)) can be
analytically extended to t = 0, which implies that X(µ), Y (µ) are analytic as functions of t =√
µ at
t = 0. QED
Theorem 3.2 is similar to Theorem 3.2 in [33], but in the latter, the sufficient condition is also
necessary for analyticity. One may ask whether the same holds here.
13
If (X1(t), Y1(t)) is analytic at t = 0, then clearly, (W1)12(t) is analytic at t = 0. What about the
condition “(W1)12(t) tends to zero as t → 0”? If this condition holds, then the sufficient condition for
asymptotic analyticity of (X1(t), Y1(t)) in the above Theorem 3.2, which is equivalent to “(W1)12(t)
converges to zero as t → 0 and is analytic at t = 0”, is also necessary.
Suppose (X1(t), Y1(t)) derived from an off-central path, (X(µ), Y (µ)), corresponding to the NT
direction is analytic at t = 0. These lead to the following conditions which must be satisfied by
X∗1 = limt→0 X1(t), Y ∗
1 = limt→0 Y1(t), W ∗1 = limt→0 W1(t):
W ∗1 Y ∗
1 W ∗1 = X∗
1 , (22)
W ∗1
(Y ∗
1 )11 0
0 −(Y ∗1 )22
W ∗
1 =
(X∗
1 )11 0
0 −(X∗1 )22
. (23)
Here, X∗1 , Y ∗
1 , W ∗1 are symmetric, positive definite matrices.
Condition (22) is derived from the fact that (X1(t), Y1(t)) is derived from an off-central path
corresponding to the NT direction, while Condition (23) is necessary for (X1(t), Y1(t)) to be analytic
at t = 0.
Also, from the monotonicity assumption of SDLCP (1)-(3), Assumption 2.1(a), we have X∗1 , Y ∗
1
must also satisfy
(X∗1 )12 • (Y ∗
1 )12 ≥ 0. (24)
Does Conditions (22)-(24) suffice to conclude that (W1)12(t) converges to zero as t → 0, that is,
(W ∗1 )12 = 0?
It turns out that in general, using Conditions (22)-(24), we cannot conclude that (W ∗1 )12 = 0. Here
is a counterexample:
Example 3.1 Let
(W ∗1 )12 =
0 0
β 0
, (Y ∗
1 )12 =
0 β1
−β 0
,
(Y ∗1 )11 = (Y ∗
1 )22 = (W ∗1 )11 = (W ∗
1 )22 = I2×2.
Choose β be a small value between 0 and 1, and
β21 = −β4 + β2.
We have X∗1 , Y ∗
1 and W ∗1 defined this way, where X∗
1 = W ∗1 Y ∗
1 W ∗1 , are symmetric, positive definite
matrices satisfying (22)-(24) with (24) being satisfied with equality, while (W ∗1 )12 6= 0.
14
However, under a further restriction on the matrices involved, the conclusion “(W ∗1 )12 = 0” holds,
as shown in the following proposition:
Proposition 3.2 If the upper left hand block of every matrix involved is one dimensional, Conditions
(22)-(24) implies that (W ∗1 )12 = 0.
Proof. By pre-multiplying and post-multiplying Y ∗1 by
(Y ∗
1 )−1/211 0
0 (Y ∗1 )−1/2
22
, (25)
we can assume the diagonal blocks of Y ∗1 to be identity matrices. Also, in order for Conditions (22)-(24)
to hold for the new Y ∗1 , we have to take the new W ∗
1 to be W ∗1 pre- and post-multiplied by the inverse
of the above matrix (25).
With these transformations, if the upper left hand block of every matrix involved is one dimensional,
we have
Y ∗1 =
1 yT
y I
, W ∗
1 =
w11 wT
w (W ∗1 )22
,
where w11 is a scalar and y, w are column vectors.
We are done if we can show that w = 0.
From Conditions (22) and (23), we have
w11wT y + ‖w‖2 = 0, (26)
2wwT + (W ∗1 )22ywT + wyT (W ∗
1 )22 = 0, (27)
w11w = (W ∗1 )22w. (28)
Post-multiplying the equation in (27) by w, we obtain
2‖w‖2w + ‖w‖2(W ∗1 )22y + wyT (W ∗
1 )22w = 0.
Using (28) and then (26) on the above equation, we obtain y = −(W ∗1 )−1
22 w, if we assume w 6= 0.
Using Condition (24), we have
w11wT y + (wT y)2 + w11y
T (W ∗1 )22y + wT (W ∗
1 )22y ≥ 0.
15
Substituting y = −(W ∗1 )−1
22 w derived earlier into the above inequality and using (28), we obtain
‖w‖2(‖w‖2
w211
− 1)≥ 0.
Since we assume w 6= 0, we then have ‖w‖ ≥ w11. But this is a contradiction to the positive definiteness
of W ∗1 . Hence we are done. QED
From Proposition 3.2, we have the following corollary:
Corollary 3.1 Let m = 1 or m = n− 1, that is, a primal optimal solution X∗ to SDLCP (1)-(3) has
rank 1 or n− 1.
Suppose (X(µ), Y (µ)), µ > 0, is an off-central path for SDLCP (1)-(3) under Assumptions 2.1 and 3.1.
Then X(µ), Y (µ) are analytic as functions of t =√
µ at t = 0 if and only if W12(µ) converges to zero
as µ tends to zero, and is analytic as a function of t =√
µ at t = 0.
Remark 3.3 From Example 3.1, we see that for 2 ≤ m ≤ n−2, the sufficient condition in Theorem 3.2
is unlikely to be also necessary for analyticity of an off-central path corresponding to the NT direction,
w.r.t.√
µ, in general.
3.2 Analyticity w.r.t. µ.
Now, let us turn our attention to the analyticity of off-central paths, corresponding to the NT direction,
w.r.t. µ.
To analyze the analyticity of an off-central path, (X(µ), Y (µ)), with respect to µ, a necessary
condition is that X12(µ) = O(µ) and Y12(µ) = O(µ). Hence, we take X12(µ) = O(µ), Y12(µ) = O(µ)
from now onwards. Also, a necessary condition for X(µ), Y (µ) to be analytic at µ = 0 is W12(µ) =
O(√
µ). To see this, we observe that X12(µ) = O(µ), Y12(µ) = O(µ) imply that W12(µ) → 0 as µ → 0.
Also, X(µ), Y (µ) analytic at µ = 0 implies that they are analytic as a function of t =√
µ at t = 0.
That is, X1(t), Y1(t) are analytic at t = 0. Hence, W1(t), in particular, (W1)12(t) = W12(t2) is analytic
at t = 0. Therefore, with W12(µ) → 0 as µ → 0, we have W12(µ) = O(√
µ).
Analyzing analyticity w.r.t. µ involves different transformation on X(µ) and on Y (µ), as compared
to the case, ((9), (10)), of analyticity w.r.t.√
µ, [34].
Following [34], let us define X(µ) and Y (µ) which are related to X(µ), Y (µ) by
X(µ) = X(µ)
I 0
0 µI
(29)
and
Y (µ) =
µI 0
0 I
Y (µ). (30)
16
Since X11(µ) = Θ(1), X12(µ) = O(µ), X22(µ) = Θ(µ), we have X(µ) = O(1). Similarly, we have
Y (µ) = O(1).
From WY W = X which must be satisfied by (X(µ), Y (µ)) - an off-central path corresponding to
the NT direction, we have
W
µI 0
0 I
Y W = X
I 0
0 µI
which implies that
W
√
µI 0
0 1õI
Y W
√
µI 0
0 1õI
= X.
Define
W (µ) := W (µ)
√
µI 0
0 1õI
. (31)
Therefore, we have W Y W = X. Also, X(µ), Y (µ), W (µ) are invertible even in the limit as µ → 0.
Note that accumulation points of W (µ) exist as µ → 0 since W (µ) = O(1). The latter follows from
W11(µ) = O(
1õ
), W22 = O(
√µ) and W12(µ) = O(
õ).
Since the “tilde” matrices are in general nonsymmetric, instead of using the operation ⊗s and the
map svec, we consider the operation ⊗ and the map vec (with inverse mat) from now on. Properties of
these can be found for example in Chapter 4 of [8] and the appendix of [39].
From (4) (with P , such that PT P = W−1, W a symmetric, positive definite matrix and WY W =
X), we have
X ′ + WY ′W =1µ
X.
Expressing the latter in terms of X, Y , W , their derivatives and µ, we obtain
vec(X ′) + (WT ⊗ W )vec(Y ′) =1µ
vec
X
I 0
0 0
− W
I 0
0 0
Y W
. (32)
Now, in the wider context when we treat X, Y ∈ <n×n instead of in Sn, let us introduce the
following:
(Eij − Eji) ·X = 0, 1 ≤ i < j ≤ n, (33)
(Eij − Eji) · Y = 0, 1 ≤ i < j ≤ n, (34)
to (2) to form
(A1 B1)
vec(X)
vec(Y )
=
q
0
, (35)
17
where
(A1 B1) =
vec((A1)1) vec((B1)1)...
...
vec((A1)n2+n(n−1)/2) vec((B1)n2+n(n−1)/2)
:=
vec(A1)T vec(B1)T
......
vec(An)T vec(B
n)T
vec(Eij − Eji)T1≤i<j≤n 0
0 vec(Eij − Eji)T1≤i<j≤n
∈ <(n2+n(n−1)/2)×2n2.
Conditions (33), (34) are introduced to ensure that although X, Y belong to <n×n, they are
necessarily symmetric.
From (35), together with (32) (see [34]), we have the following ODE system: A1(µ) B1(µ)
I WT ⊗ W
vec(X ′)
vec(Y ′)
=
1µ
−A1(µ)
0 0
0 I
⊗ I
vec(X)− B1(µ)
I ⊗
I 0
0 0
vec(Y )
vec
X
I 0
0 0
− W
I 0
0 0
Y W
,
(36)
where
A1(µ)k· :=
vec
((A1)k)11 µ((A1)k)12
((A1)k)21 µ((A1)k)22
T
for 1 ≤ k ≤ i′1
vec
0 ((A1)k)12
0 ((A1)k)22
T
for i′1 + 1 ≤ k ≤ n2 + n(n− 1)/2
(37)
and
B1(µ)k· :=
vec
µ((B1)k)11 µ((B1)k)12
((B1)k)21 ((B1)k)22
T
for 1 ≤ k ≤ i′1
vec
((B1)k)11 ((B1)k)12
0 0
T
for i′1 + 1 ≤ k ≤ n2 + n(n− 1)/2
, (38)
for some fixed i′1.
Details on how A1(µ),B1(µ) are derived and their properties can be obtained from [34].
18
The analyticity of the matrix A1(µ) B1(µ)
I WT ⊗ W
,
which appears on the left hand side of (36), in an open neighborhood of (µ, X(µ), Y (µ)), µ > 0, of an
off-central path, and its accumulation points, is given in the following proposition. Analyticity of this
matrix is required to analyze the asymptotic analyticity behavior of paths corresponding to the NT
direction, using ODE system (36) (or (42)).
Before we state and prove the proposition, let us first state a well-known result which will be used
in the proof of the proposition.
Result 3.1 Let A ∈ <p×p and B ∈ <q×q. The equation AX + XB = C has an unique solution
X ∈ <p×q for each C ∈ <p×q if and only if σ(A) ∩ σ(−B) = ∅.Here, σ(A) refers to the set of all eigenvalues of the square matrix A.
The above result can be found for example in Chapter 4 of [8].
Proposition 3.3 There exists an open neighborhood U ⊂ <×<n×n×<n×n of (µ, X(µ), Y (µ)), µ > 0,
of an off-central path, and (0, X∗, Y ∗) corresponding to any accumulation point of the off-central path,
such that for (µ, X, Y ) ∈ U , the matrix A1(µ) B1(µ)
I WT ⊗ W
is analytic at (µ, X, Y ), where W exists, is an analytic function of (X, Y ), and satisfies W Y W = X.
Proof. Consider
f : <n×n ×<n×n ×<n×n −→ <n×n
defined by
f(X, Y , W ) = W Y W − X.
We have
∇W
f(X, Y , W )U = W Y U + UY W , U ∈ <n×n.
We need only show that∇W
f(X, Y , W ) is invertible for (X, Y , W ) with W Y W = X and (X, Y ) derived
from the set of (µ, X(µ), Y (µ)), µ > 0, of an off-central path, or any of its accumulation points. The
proposition then follows from the Implicit Function Theorem.
19
First, we consider (X, Y ) to be derived from the set (µ, X(µ), Y (µ));µ > 0.
From (30) and (31), we have
W Y =1õ
WY
and
Y W =1õ
1õI 0
0√
µI
Y W
√
µI 0
0 1õI
.
Since W and Y are symmetric and positive definite, we see from the above relations that σ(W Y ) ∩σ(−Y W ) = ∅. Hence, by Result 3.1, ∇
Wf(X, Y , W ) is invertible.
Let (X, Y ) be derived from one of the accumulation points of (µ, X(µ), Y (µ)) as µ tends to zero, with
corresponding W . Let us denote them by X∗, Y ∗ and W ∗ respectively.
Let (Xk, Yk, Wk) be derived from an off-central path corresponding to µk > 0, with limk→∞ µk = 0,
limk→∞ Xk = X∗, limk→∞ Yk = Y ∗ and limk→∞ Wk = W ∗. (Actually, we should have written Xk =
X(µk), Yk = Y (µk), Wk = W (µk). They are written this way for the sake of simplification of notations.)
We have WkYk = 1õk
WkYk by (30) and (31), where again Wk and Yk stand for W (µk) and Y (µk)
respectively.
Now,
Wk =√
µk
1õk
I 0
0 I
(Wk)11
õk(Wk)12
õk(Wk)T
12 (Wk)22
1õk
I 0
0 I
,
where (Wk)11, (Wk)22 are symmetric, positive definite matrices even in the limit as k → ∞, (Wk)11 =
Θ(1), (Wk)22 = Θ(1) and (Wk)12 = O(1), and
Yk =
√
µkI 0
0 I
(Yk)11
õk(Yk)12
õk(Yk)T
12 (Yk)22
√
µkI 0
0 I
,
where (Yk)11, (Yk)22 are symmetric, positive definite matrices even in the limit as k → ∞, (Yk)11 =
Θ(1), (Yk)22 = Θ(1) and (Yk)12 = O(1).
The above properties on (Wk)ij , (Yk)ij hold due to (Wk)11 = Θ(
1õk
), (Wk)22 = Θ(
õk), (Wk)12 =
O(√
µk), and (Yk)11 = Θ(µk), (Yk)22 = Θ(1), (Yk)12 = O(µk).
Note that the limit of each “check” submatrix exists as k tends to ∞. (If not, then they exist by taking
a suitable subsequence.)
20
Therefore,
WkYk =
(Wk)11(Yk)11 + µk(Wk)12(Yk)T
12 (Wk)11(Yk)12 + (Wk)12(Yk)22
µk(Wk)T12(Yk)11 + µk(Wk)22(Yk)T
12 µk(Wk)T12(Yk)12 + (Wk)22(Yk)22
.
Letting µk → 0, we have
W ∗Y ∗ =
W ∗
11Y∗11 W ∗
11Y∗12 + W ∗
12Y∗22
0 W ∗22Y
∗22
.
Similarly,
Y ∗W ∗ =
Y ∗
11W∗11 Y ∗
11W∗12 + Y ∗
12W∗22
0 Y ∗22W
∗22
.
Therefore, σ(W ∗Y ∗) ∩ σ(−Y ∗W ∗) = ∅. And ∇W
f(X∗, Y ∗, W ∗) is invertible, by Result 3.1 again.
Hence, we are done. QED
Note that the matrix A1(µ) B1(µ)
I WT ⊗ W
is not square. However, we show in the following proposition that it is one-to-one in an open neighbor-
hood of (µ, X(µ), Y (µ)), µ > 0, of an off-central path, and any of its accumulation points.
Proposition 3.4 There exists an open neighborhood U ⊂ <×<n×n×<n×n of (µ, X(µ), Y (µ)), µ > 0,
of an off-central path, and (0, X∗, Y ∗) corresponding to any accumulation point of the off-central path,
such that the matrix A1(µ) B1(µ)
I WT ⊗ W
is one-to-one in U , where W exists, is unique, and is such that W Y W = X, for (µ, X, Y ) ∈ U .
Proof. For µ > 0, we need only show that for X1 = X1(µ) and Y1 = Y1(µ), where (X1(µ), Y1(µ)) is
derived from an off-central path, if A1(µ) B1(µ)
I WT ⊗ W
vec(U)
vec(V )
= 0, for U, V ∈ <n×n, (39)
then U = V = 0.
From A1(µ)vec(U) + B1(µ)vec(V ) = 0 in (39), using monotonicity, we have vec(V T )T vec(U) ≥ 0.
Therefore, using vec(U) + (WT ⊗ W )vec(V ) = 0 in (39), we have vec(V T )T (WT ⊗ W )vec(V ) ≤ 0.
21
Let us now look at the expression vec(V T )T (WT ⊗ W )vec(V ). We have
vec(V T )T (WT ⊗ W )vec(V )
= Tr(V WV W )
=1µ
Tr
µI 0
0 I
V W
I 0
0 µI
µI 0
0 I
V W
1µ 0
0 I
=1µ
Tr
µI 0
0 I
V W
1õI 0
0√
µI
µI 0
0 I
V W
1õ 0
0√
µI
=1µ
∥∥∥∥∥∥∥
W
1õ 0
0√
µI
1/2 µI 0
0 I
V
W
1õ 0
0√
µI
1/2∥∥∥∥∥∥∥
2
≥ 0,
since W
1õ 0
0√
µI
is symmetric, positive definite, and
µI 0
0 I
V is symmetric.
Hence, we conclude that U = V = 0.
Now, suppose (X∗, Y ∗) is an accumulation point of (X(µ), Y (µ)) as µ → 0. We have W ∗ is such that
W ∗Y ∗W ∗ = X∗. Let us consider A1(0) B1(0)
I (W ∗)T ⊗ W ∗
vec(U0)
vec(V0)
= 0, for U0, V0 ∈ <n×n. (40)
Again, we need only show that U0 = V0 = 0.
By monotonicity, we have using A1(0)vec(U0) + B1(0)vec(V0) = 0 in (40) that vec(V T0 )T vec(U0) ≥ 0.
Also, we can show that (V0)11 = (V0)T11, (V0)22 = (V0)T
22 and (V0)21 = 0. (See proof of Proposition 2.2
in [34] for details.)
Note that W ∗21 = 0, and W ∗
11, W∗22 are symmetric, positive definite. Together with the above properties
of V0, we have
vec(V T0 )T ((W ∗)T ⊗ W ∗)vec(V0)
= Tr(V0W∗V0W
∗)
= Tr
(V0)11 (V0)12
0 (V0)22
W ∗
11 W ∗12
0 W ∗22
×
(V0)11 (V0)12
0 (V0)22
W ∗
11 W ∗12
0 W ∗22
= Tr((V0)11W ∗11(V0)11W ∗
11) + Tr((V0)22W ∗22(V0)22W ∗
22)
= ‖(W ∗11)
1/2(V0)11(W ∗11)
1/2‖+ ‖(W ∗22)
1/2(V0)22(W ∗22)
1/2‖ ≥ 0.
22
Since vec(V T0 )T vec(U0) ≥ 0, we also have ‖(W ∗
11)1/2(V0)11(W ∗
11)1/2‖ + ‖(W ∗
22)1/2(V0)22(W ∗
22)1/2‖ ≤ 0.
Hence, (V0)11, (V0)22 = 0.
Using vec(U0) + ((W ∗)T ⊗ W ∗)vec(V0) = 0 in (40), we deduce from what is known of V0 and W ∗, that
(U0)11 = 0, (U0)22 = 0 and (U0)21 = 0. Also,
(U0)12 + W ∗11(V0)12W ∗
22 = 0. (41)
By monotonicity arguments, we have Tr((V0)T12(U0)12) ≥ 0 (see proof of Proposition 2.2 in [34]). Hence,
using (41), we conclude that (U0)12 = (V0)12 = 0. QED
As in [34], observe that we can further reduce the number of rows in the above ODE system (36)
to form the following ODE system: A1(µ) B1(µ)
I
WT ⊗ W
vec(X ′)
vec(Y ′)
=
1µ
−A1(µ)
0 0
0 I
⊗ I
vec(X)− B1(µ)
I ⊗
I 0
0 0
vec(Y )
vec
X
I 0
0 0
− W
I 0
0 0
Y W
,
(42)
where the “hat” operation is defined as follows:
Definition 3.1 Given D ∈ <n2×n2, define D ∈ <(n2−m(n−m))×n2
such that for all u ∈ <n2, Du
consists of entries of V11, V12, V22 in the order in which they appear in the image of Du, where Du =
vec
V11 V12
V21 V22
.
Also, define the map vec from <n×n to <n2−m(n−m) such that, given V ∈ <n×n, vec(V ) is the vector
in <n2−m(n−m) obtained from vec(V ) by removing entries of V21.
Remark 3.4 The matrix on the left hand side of the above ODE system (42) is again one-to-one
and analytic in an open neighborhood of (µ, X(µ), Y (µ)), µ > 0, of an off-central path, and any of its
accumulation points.
Using the ODE system (42) (or (36)), we have the following theorem, which is similar to Theorem
2.1 in [34]:
Theorem 3.3 Let (X(µ), Y (µ)), µ > 0, be an off-central path for SDLCP (1)-(3) under Assumptions
2.1 and 3.1. Then X(µ), Y (µ) are analytic at µ = 0 if and only if X12(µ) = O(µ), Y12(µ) = O(µ),
W12(µ) = O(√
µ), and 1µ (W11Y11W12 + W11Y12W22)(µ) → 0 as µ → 0 and is analytic at µ = 0.
23
Proof. First, observe that “ 1µ (W11Y11W12 + W11Y12W22)(µ) → 0 as µ → 0 and is analytic at µ = 0” is
equivalent to “(W11Y11W12 + W11Y12W22)(µ) → 0 as µ → 0 and is analytic at µ = 0”. Also, X12(µ) =
O(µ) ⇔ X12(µ) = O(1), Y12(µ) = O(µ) ⇔ Y12(µ) = O(1) and W12(µ) = O(√
µ) ⇔ W12(µ) = O(1).
(⇒) Suppose X(µ), Y (µ) are analytic at µ = 0.
Then it is clear that X12(µ) = O(µ) and Y12(µ) = O(µ), and W12(µ) = O(√
µ).
Also, since the left hand side of the ODE system (42) is analytic at µ = 0, and that
W11Y11W12 + W11Y12W22 =
X
I 0
0 0
− W
I 0
0 0
Y W
12
,
we have the required result.
(⇐) Suppose X12(µ) = O(1), Y12(µ) = O(1), W12(µ) = O(1), and (W11Y11W12 + W11Y12W22)(µ) → 0
as µ → 0 and is analytic at µ = 0.
We have (X(µ), Y (µ), W (µ)) satisfies W Y W = X, X21 = µXT12, Y21 = µY T
12, W21 = µWT12 and
X
I 0
0 0
− W
I 0
0 0
Y W
=
X11 − W11Y11W11 − W11Y12W21 −W11Y11W12 − W11Y12W22
X21 − W21Y11W11 − W21Y12W21 −W21Y11W12 − W21Y12W22
.
These imply that for X = X(µ), Y = Y (µ),
vec
X
I 0
0 0
− W
I 0
0 0
Y W
in (42) can be written as a product of µ, and an analytic function of µ, X, Y at any accumulation points
of (µ, X(µ), Y (µ)), derived from the given off-central path, as µ tends to zero.
Also, for X = X(µ), Y = Y (µ),
−A1(µ)
0 0
0 I
⊗ I
vec(X)− B1(µ)
I ⊗
I 0
0 0
vec(Y )
in (42) can be written as a product of µ and an analytic function of X, Y (see [34]).
Together with Remark 3.4, we see that (X(µ), Y (µ)) of the given off-central path satisfies an ODE
system in standard form (y′ = f(t, y)), derived from (42), whose right hand side is analytic at any
accumulation points of (µ, X(µ), Y (µ)). Hence, by Theorem 4.1 of [4], pp. 15 and Theorem 2.1 of [32],
(X(µ), Y (µ)) can be analytically extended to µ = 0, which implies that the same holds for (X(µ), Y (µ)),
due to (29), (30). And we are done. QED
24
4 Conclusion.
In this paper, we provide a few off-central path examples, corresponding to the NT direction, for a
particularly simple and nice SDP, whose behavior near the unique solution of this SDP is not well-
behaved. We also provide in Section 3 necessary and sufficient conditions for when an off-central path,
corresponding to the NT direction, for a general SDLCP is analytic, first w.r.t.√
µ and then w.r.t. µ,
at µ = 0. In the analysis in Section 3, we follow the approach in [33,34]. Similar analysis like in [32]
for the NT case for the simple SDP example in Section 2 proves to be too difficult for this paper.
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